- #1

Math Amateur

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I need help in order to fully understand the set of Borel sets ... ...

The relevant text reads as follows:

My questions related to the above text are as follows:

__QUESTION 1__In the above text by Axler we read the following:

" ... ... However, the set of all such intersections is not the set of Borel sets (because it is not closed under countable unions). ... ..."Can someone please explain why exactly that the set of all such intersections is not the set of Borel sets ... ? Why exactly is such a set not closed under countable unions and why is this relevant?

__QUESTION 2__In the above text by Axler we read the following:

" ... ... The set of all countable unions of countable intersections of open subsets of $\mathbb{R}$ is also not the set of Borel sets (because it is not closed under countable intersections). ... ... "Can someone please explain why exactly that the set of all countable unions of countable intersections of open subsets of $\mathbb{R}$ is not the set of Borel sets ... ? Why exactly is such a set not closed under countable intersections and why is this relevant?

Help with the above two questions will be much appreciated ...

Peter